Exercises

1. Differentiate each of the following functions.

   `ftext[(]t text[)]=e^(3t)` `gtext[(]t text[)]=e^(-2t)` `Ptext[(]t text[)]=1.2^t` `htext[(]t text[)]=3t-5t^2` `phitext[(]t text[)]=6-e^(-3t)` `gtext[(]t text[)]=2t+e^(-t)` `Ftext[(]t text[)]=e^(-3t)+6t^2`

   `Gtext[(]t text[)]=6-e^(-2)` `ptext[(]t text[)]=2^(3t)` `Htext[(]t text[)]=3+t+5t^2` `psitext[(]t text[)]=e^(-3t)-6t+7` `rtext[(]t text[)]=2t^3-e^(-t)` `Qtext[(]t text[)]=7t^4+5t^3-4t^2+3t-10` `wtext[(]t text[)]=t^3+sqrt(2)e^(2t)`

2. Solve each of the following initial value problems.

   `(dy)/(dt)=1-2t` with `y=3` at `t=0` `(dy)/(dt)=-4-2y` with `y=1` at `t=0` `(dy)/(dt)=3t^2` with `y=5` at `t=2` `(dz)/(dt)=2-1.7e^(-1.7t)` with `z=1` at `t=0` `(dw)/(dt)=-1.7e^(-1.7t)` with `w=3` at `t=0` `(dY)/(dt)=2t+2e^(-2t)` with `Y=3` at `t=0` `(dP)/(dt)=2t-3e^(-3t)` with `P=1` at `t=0` `(dq)/(dt)=1+3t^2+3^(-t)` with `q=1` at `t=0` `(du)/(dt)=8t^3-text[(]ln 2 text[)]2^t` with `u=3` at `t=0` `(dx)/(dt)=-21e^(-3t)-6t+3t^2` with `x=3` at `t=0`


3. Suppose the marble is dropped from a height of 100 feet (approximately the height of a 10-story building).

   1. What is its velocity (in feet per second) just before impact?
   2. How long does it take to hit the ground?
4. Suppose the marble is dropped from a height of 4 feet (i.e., from arm's height above the ground).

   1. What is its velocity (in feet per second) just before impact?
   2. How long does it take to hit the ground?
5. 1. Suppose the marble is tossed straight up from a height of 535 feet with an initial speed of 40 feet per second. How does this affect the time of fall and velocity at impact?
   2. Suppose it is tossed straight down at 40 feet per second. How does that affect the time of fall and velocity at impact?
6. Suppose a baseball is thrown straight up, from a height of 5 feet above the ground, with an initial velocity of 70 feet per second. Ignoring air resistance and other nongravitational forces, find formulas for its velocity and height as functions of time, and then answer the following questions, not necessarily in the order given.

   1. How high does the ball rise?
   2. How fast is it going at the peak of its flight?
   3. How long does it take to reach its peak height?
   4. How fast is the ball going when it hits the ground?
   5. What is the total elapsed time from release of the ball until it hits the ground?
7. Suppose a marble is thrown downward with a velocity of 10 feet per second from a height of 535 feet. How long will it take to hit the ground? How fast will it be traveling when it reaches the ground?
8. Suppose your car is capable of accelerating smoothly from a standing start at the rate of `1.4` feet per second per second.

   1. Find the acceleration, the velocity, and the distance traveled, all as functions of time, over the first minute of travel.
   2. What is the speedometer reading at the end of one minute?
   3. How far, in miles, does the car travel in that time?
9. In Exercise 5 you modified the falling marble problem by giving the marble an initial velocity of 40 feet per second (up or down).
   1. Solve the problem for an arbitrary initial velocity `v_0` (positive or negative) and an arbitrary starting height `s_0`.
   2. What are the time of fall and the velocity at impact? (Your answer will involve `v_0` and `s_0`.)
10. Although many people know of the famous golf ball experiment, conducted during the Apollo 11 mission to the moon, most people are unfamiliar with the experiments performed by the Frenchman, Forget Menot. Having calculated the acceleration due to gravitational force on the moon to be `5.4` feet per second per second, Forget threw a ball straight up with an initial velocity of `5` feet per second. Unfortunately, Forget neglected to bring his watch and was unable to estimate the total time it took for the ball to rise to its peak and return to the ground. Help Capitaine Menot by calculating this time and also by calculating the height of the ball at its peak.

    Problem contributed by Sam Morris, Duke University.

11. At a certain instant, just before lifting off, a plane is traveling down the runway at `285` kilometers per hour. The pilot suddenly realizes something is wrong and aborts the takeoff by cutting power and applying the brakes. Assume that the effect of this action is a deceleration proportional to time `t`. After `28` seconds the plane comes to a stop. How far does the plane travel after the takeoff is aborted?